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Abstract:  We present our recent extension of Allard's celebrated rectifiability theorem  to the setting of  varifolds  with locally bounded  first variation with respect to an anisotropic integrand. In particular, we identify a necessary and sufficient condition on the integrand to obtain the rectifiability of every (d)-dimensional varifold with locally bounded first variation and positive (d)-dimensional density.  In codimension one, this condition is shown to be equivalent to  the strict convexity  of the integrand with respect to the tangent plane.

We can apply this result to the minimization of anisotropic energies among families of \(d\)-rectifiable closed subsets of \(R^n\), closed under Lipschitz deformations (in any dimension and codimension). Easy corollaries of this compactness result are the solutions to three formulations of the Plateau problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David.

Moreover, we apply the rectifiability theorem to the energy minimization in classes of varifolds and to a compactness result of integral varifolds in the anisotropic setting.